Question

Which statement is true about the equation?

fraction 3 over 4z − fraction 1 over 4z + 3 = fraction 2 over 4z + 5

Answer 1

Interpreting the equation as 3/(4z) − 1/(4z+3) = 2/(4z+5), the domain excludes 4z = 0, −3, −5, i.e. z ≠ 0, −3/4, −5/4.

Solve: let t = 4z. Then
3/t − 1/(t+3) = 2/(t+5)
Multiply through by t(t+3)(t+5) and simplify to get 13t + 45 = 0, so t = −45/13, hence z = t/4 = −45/52.

Thus the equation has exactly one solution: z = −45/52 (which is allowed by the domain).